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We introduce a novel construction which allows us to identify the elements of the skeletons of a CW-complex $P(m)$ and the monomials in $m$ variables. From this, we infer that there is a bijection between finite CW-subcomplexes of $P(m)$, which are quotients of finite simplicial complexes, and some bigraded standard Artinian Gorenstein algebras, generalizing previous constructions in \cite{F:S}, \cite{CGIM} and \cite{G:Z}. We apply this to a generalization of Nagata idealization for level algebras. These algebras are standard graded Artinian algebras whose Macaulay dual generator is given explicitly as a bigraded polynomial of bidegree $(1,d)$. We consider the algebra associated to polynomials of the same type of bidegree $(d_1,d_2)$., 19 pages, 2 figures, AMS-LaTeX. To be published in Advances in Applied Mathematics

It is shown that not every metrizable compactum can be written as the inverse limit of an inverse sequence of finite triangulated polyhedra with simplicial bonding maps. This result came from a correspondence between Sibe Mardesic and Leonard R. Rubin, who has provided the Introduction below, and who with some suggestions from Sime Ungar and Vera Tonic, has edited the proof that was communicated to him by Sibe Mardesic. It will be found as Corollary 2.2 .

We show that the Pr\"ufer surface, which is a separable non-metrizable 2-manifold, has not the homotopy type of a CW-complex. This will follow easily from J. H. C. Whitehead's result: if one has a good approximation of an arbitrary space by a CW-complex, which fails to be a homotopy equivalence, then the given space is not homotopy equivalent to a CW-complex., Comment: 5 pages, 3 figures

We present new, unified proofs for the cell-like, $\mathbb{Z}/p$-, and $\mathbb{Q}$-resolution theorems. Our arguments employ extensions that are much simpler then those used by our predecessors. The techniques allow us to solve problems involving cohomology groups by converting them into problems about homology groups. We provide a coordinated general topological method for constructing the maps needed to witness the resolution theorems simultaneously., Comment: The newest version of the paper (v2) is considerably different from the first version (v1), including a change in the title, with errors from v1 corrected. 24 pages

We prove the following Theorem: Let X be a nonempty compact metrizable space, let $l_1 \leq l_2 \leq...$ be a sequence of natural numbers, and let $X_1 \subset X_2 \subset...$ be a sequence of nonempty closed subspaces of X such that for each k in N, $dim_{Z/p} X_k \leq l_k < \infty$. Then there exists a compact metrizable space Z, having closed subspaces $Z_1 \subset Z_2 \subset...$, and a surjective cell-like map $\pi: Z \to X$, such that for each k in N, (a) $dim Z_k \leq l_k$, (b) $\pi (Z_k) = X_k$, and (c) $\pi | {Z_k}: Z_k \to X_k$ is a Z/p-acyclic map. Moreover, there is a sequence $A_1 \subset A_2 \subset...$ of closed subspaces of Z, such that for each k, $dim A_k \leq l_k$, $\pi|{A_k}: A_k\to X$ is surjective, and for k in N, $Z_k\subset A_k$ and $\pi|{A_k}: A_k\to X$ is a UV^{l_k-1}-map. It is not required that X be the union of all X_k, nor that Z be the union of all Z_k. This result generalizes the Z/p-resolution theorem of A. Dranishnikov, and runs parallel to a similar theorem of S. Ageev, R. Jim\'enez, and L. Rubin, who studied the situation where the group was Z., Comment: 18 pages, title change in version 3, old title: "Z/p-acyclic resolutions in the strongly countable Z/p-dimensional case"

Let Z, H be spaces. In previous work, we introduced the direct system A induced by the set of maps between the spaces Z and H. Now we will consider the case that A is induced by possibly a proper subset of the maps of Z to H. Our objective is to explore conditions under which X = dirlim A will be T1, Hausdorff, regular, completely regular, pseudo-compact, normal, an absolute co-extensor for some space K, or will enjoy some combination of these properties.

In some previous paper D. Kolodziejczyk (2001) [19] we proved that there exists a finite polyhedron P with infinitely many left neighbors in the homotopy category, i.e. P homotopy dominates infinitely many finite polyhedra P i of different homotopy types and there isnʼt any homotopy type between P and P i . This answers a question of K. Borsuk (Borsuk, 1975 [2] ). The dimension of that P is 3. Here we show that there exists a finite 2-dimensional polyhedron with the same property. Some remarks on constructing examples of 2-dimensional finite polyhedra dominating infinitely many different homotopy types are also included.

As the main results of this paper we prove that for every polyhedron P with abelian or torsion-free nilpotent fundamental group there are only finitely many different homotopy types of X i such that X i × S 1 ≃ P . The same holds for any finite K ( G , 1 ) with nilpotent fundamental group in place of S 1 . The problem, if there exists a polyhedron with infinitely many direct factors of different homotopy types (K. Borsuk, 1970) [2] is still unsolved, even if we assume the second factor to be S 1 .

We answer a question of K. Borsuk (1972) showing that there exists a connected polyhedron of dimension 4 with two different decompositions into a direct product of prime factors (in the homotopy or shape category).

Suppose that K is a CW-complex. When we say that a space Y is an absolute co-extensor for K , we mean that K is an absolute extensor for Y , i.e., that for every closed subset A of Y and any map f : A → K , there exists a map F : Y → K that extends f . Our main theorem will provide several statements that are equivalent to the condition that whenever K is a CW-complex and X is a space which is the topological sum of a countable collection of compact metrizable spaces each of which is an absolute co-extensor for K , then the Stone-Cech compactification of X is an absolute co-extensor for K .

In the previous papers, in connection with a question of K. Borsuk, we proved that there exist polyhedra with polycyclic fundamental groups homotopy dominating infinitely many different homotopy types. Here we consider a few problems of K. Borsuk concerning infinite chains of polyhedra or FANR's ordered by the relation of domination (in homotopy or shape category) and obtain that for polyhedra with polycyclic-by-finite fundamental groups, there are no pathology similar to the above.

In 1968 K. Borsuk asked: Is it true that every finite polyhedron dominates only finitely many different shapes? In this question the notions of shape and shape domination can be replaced by the notions of homotopy type and homotopy domination. We obtained earlier a negative answer to the Borsuk question and next results that the examples of such polyhedra are not rare. In particular, there exist polyhedra with nilpotent fundamental groups dominating infinitely many different homotopy types. On the other hand, we proved that every polyhedron with finite fundamental group dominates only finitely many different homotopy types. Here we obtain next positive results that the same is true for some classes of polyhedra with Abelian fundamental groups and for nilpotent polyhedra. Therefore we also get that every finitely generated, nilpotent torsion-free group has only finitely many r -images up to isomorphism.

Let G be a finite group one of the forms given in the title with period 2 d and X ( n ) an n -dimensional CW-complex with the homotopy type of an n -sphere. We study the automorphism group Aut( G ) to compute the number of distinct homotopy types of spherical space forms with respect to G -actions on all CW-complexes X (2 dn −1). At the end, the groups E (X(2dn−1)/γ) of self homotopy equivalences of space forms X (2 dn −1)/ γ associated with free and cellular G -actions γ on X (2 dn −1) are determined.

Let X p → X be a covering projection. We use p ♯ :C(Y, X )→C(Y,X) to denote the map defined by p ♯ ( f )=p∘ f , where spaces of continuous maps are endowed with the compact-open topology. We prove that if Y is Hausdorff and contractible, or if Y is either a compact CW-complex or a graph with finitely many components and p ♯ is onto, then p ♯ is a covering projection.

This article explains and extends semialgebraic homotopy theory (developed by H. Delfs and M. Knebusch) to o-minimal homotopy theory (over a field). The homotopy category of definable CW-complexes is equivalent to the homotopy category of topological CW-complexes (with continuous mappings). If the theory of the o-minimal expansion of a field is bounded, then these categories are equivalent to the homotopy category of weakly definable spaces. Similar facts hold for decreasing systems of spaces. As a result, generalized homology and cohomology theories on pointed weak polytopes uniquely correspond (up to an isomorphism) to the known topological generalized homology and cohomology theories on pointed CW-complexes., Comment: To appear in Rocky Mountain Journal of Mathematics

Using algebraic topology, we find out the number of all non-parallels-structures which ann-dimensional Euclidean spaceEnadmits. The obtaining results are generalized on a manifoldMwhich isCW-complex.

In the paper titled “Bockstein basis and resolution theorems in extension theory” (Tonic, 2010 [10] ), we stated a theorem that we claimed to be a generalization of the Edwards–Walsh resolution theorem. The goal of this note is to show that the main theorem from Tonic (2010) [10] is in fact equivalent to the Edwards–Walsh resolution theorem, and also that it can be proven without using Edwards–Walsh complexes. We conclude that the Edwards–Walsh resolution theorem can be proven without using Edwards–Walsh complexes.

We prove a generalization of the Edwards-Walsh Resolution Theorem: Theorem: Let G be an abelian group for which $P_G$ equals the set of all primes $\mathbb{P}$, where $P_G=\{p \in \mathbb{P}: \Z_{(p)}\in$ Bockstein Basis $ \sigma(G)\}$. Let n in N and let K be a connected CW-complex with $\pi_n(K)\cong G$, $\pi_k(K)\cong 0$ for $0\leq k< n$. Then for every compact metrizable space X with $X\tau K$ (i.e., with $K$ an absolute extensor for $X$), there exists a compact metrizable space Z and a surjective map $\pi: Z \to X$ such that (a) $\pi$ is cell-like, (b) $\dim Z \leq n$, and (c) $Z\tau K$., Comment: 23 pages

We prove the existence of extension dimension for a much expanded class of spaces. First we obtain several theorems which state conditions on a polyhedron or $\mathop{\mathrm{CW}}$ -complex $K$ and a space $X$ in order that $X$ be an absolute co-extensor for $K$ . Then we prove the existence of and describe a wedge representative of extension dimension for spaces in a wide class relative to polyhedra or $\mathop{\mathrm{CW}}$ -complexes. We also obtain a result on the existence of a “countable” representative of the extension dimension of a Hausdorff compactum.

Ein Arrangement von Hyperebenen A (kurz: Arrangement) besteht aus einer endlichen Menge von linearen Unterr¨aumen der Kodimension 1 in einem endlichdimensionalen Vektorraum. Jede dieser Hyperebenen H ist der Kern einer linearen Abbildung αH, welche bis auf Konstanten eindeutig ist. AR n−1 bezeichnet das Braid-Arrangement im Rn. Es besteht aus den Hyperebenen Hi,j := {x ∈ Rn | xi = xj} f¨ur 1 ≤ i < j ≤ n. BR n bezeichnet das Arrangement im Rn, dass zus¨atzlich zu den Hyperebenen Hi,j des Braid-Arrangements, die Hyperebenen Hi,−j := {x ∈ Rn | xi = −xj}, 1 ≤ i < j ≤ n, und die Koordinaten-Hyperebenen Hi := {x ∈ Rn | xi = 0}, i = 1, . . . , n, enth¨alt. Als Komplexifizierung eines reellen Arrangments im Rn versteht man das Arrangement im Cn, welches durch dieselben definierenden Linearformen gegeben ist. Wir verzichten auf den Index C und bezeichnen mit An−1 bzw. Bn die Komplexifizierungen der beiden Arrangements, die wir oben definiert haben. Die Notation ist in Anlehnung an die zugehörigen Spiegelungsgruppen An−1 und Bn gewählt. Für ein Hyperebenen Arrangement A definieren wir das Komplement M(A) von A als das Komplement der Vereinigung aller Hyperebenen in A. In dieser Arbeit untersuchen wir die Komplemente M(An−1) ⊂ Cn und M(Bn) ⊂ Cn der Komplexifizierungen der beiden obigen reellen Arrangements. Die topologischen Eigenschaften solcher Komplemente werden seit den frühen 1970er Jahren untersucht. P. Deligne zeigte 1972, dass das Komplement eines komplexifizierten Arrangements K(π, 1) ist, falls die Regionen, die durch die 103 Hyperebenen im Rn entstehen simpliziale Kegel sind [7]. Wichtig f¨ur diese Arbeit ist die Tatsache, dass das Komplement eines komplexifizierten reellen Arrangements homotopie-äquivalent zu einem regulären CW-Komplex ist. Dies wurde 1987 von M. Salvetti gezeigt [18]. Die Gruppen Hi(Xi,Xi−1) des zellulären Ko-Kettenkomplexes eines CW-Komplexes X sind frei abelsch, mit den i-Zellen von X als Basis. Daher nennen wir einen CW-Komplex minimal, falls die Anzahl der i-Zellen genau dem Rang der Gruppe Hi(X,Q) entspricht. Ausgehend von regulären CW-Komplexen, wie sie Salvetti beschreibt, konstruieren wir minimale Komplexe An−1 und Bn für die Komplemente M(An−1) und M(Bn). Dass heißt, wir konstruieren jeweils einen CW-Komplex, der homotopie-äquivalent zu M(An−1) bzw. M(Bn) ist und der eine minimale Anzahl von Zellen besitzt. Hierzu benutzen wir die Methoden der diskreten Morse Theorie. Diese wurde in den späten 1990er Jahren von R. Forman entwickelt [8]. Mit diesen Methoden kann man die Anzahl der Zellen eines regulären CW-Komplexes verkleinern, ohne seinen Homotopie-Typ zu verändern. Parallel zu unserer Arbeit wurde ein allgemeiner Ansatz untersucht, wie man CW-Komplexe mit Hilfe diskreter Morse Theorie findet, welche homotopie- äquivalent zum Komplement eines gegebenen Arrangements sind [19]. Unser Ansatz unterscheidet sich von jenem und führt, im Falle unserer Beispiele, zu detaillierteren Resultaten. Es ist bekannt, dass die Erzeuger der Kohomologie-Gruppen der Komplemente M(An−1) und M(Bn−1) den Elementen der zugeh¨origen Spiegelungsgruppen Sn und SB n entsprechen [1]. Hierbei ist Sn die symmetrische Gruppe und SB n die Gruppe der signed permutations. Diese besteht aus den Permuationen der Menge [±n] := {1, . . . , n,−n, . . . ,−1}, so dass ω(−a) = −ω(a) für alle a ∈ [±n]. Tatsächlich ist die Anzahl der Zellen der minimalen Komplexe An−1 und Bn gleich der Anzahl der Elemente der Gruppen Sn bzw. SB n . Die Zell-Ordnung eines CW-Komplexes X ist definiert durch die Ordnung auf den Zellen von X, mit σ ≤ τ für zwei Zellen σ und τ von X, genau dann, wenn der Abschluss von σ im Abschluss von τ enthalten ist. Die partiell geordnete Menge der auf diese Weise geordneten Zelle von X heißt face poset von X. Ein großer Teil der Arbeit befasst sich mit der Zell-Ordnung der beiden minimalen Komplexe. Im Falle des Komplexes An−1 lässt sich eine prägnante Beschreibung der Ordnung herleiten. Die Zellordnung des Komplexes Bn scheint zu kompliziert, um ebenso prägnant beschrieben zu werden. Daher verfolgen wir den Ansatz, diese mit Hilfe bestimmter Mechanismen zu beschreiben, welche auf die Zellen des Komplexes angewendet werden können, um neue Zellen zu erzeugen., An arrangement of hyperplanes (or just an arrangement) A is a finite collection of linear subspaces of codimension 1 in a finite dimensional vector space. Each hyperplane H is the kernel of a linear function αH, which is unique up to a constant. ARn−1 denotes the braid arrangement in Rn, consisting of the hyperplanes Hi,j := {x ∈ Rn | xi = xj}, for 1 ≤ i < j ≤ n. BR n denotes the arrangement in Rn which in addition to the hyperplanes Hi,j of the braid arrangement consists of the hyperplanes Hi,−j := {x ∈ Rn | xi = −xj}, for 1 ≤ i < j ≤ n and the coordinatehyperplanes Hi := {x ∈ Rn | xi = 0}, for i = 1, . . . , n. A complexification of a real hyperplane arrangement in Rn is defined to be the hyperplane arrangement in Cn which is defined by the same linear forms. We omit the index C and denote by An−1 and Bn the complexifications of the real arrangements AR n−1 and BR n, respectively. The notation is chosen according to the respective reflection groups of type An−1 and Bn. For an arrangement of hyperplanes A we denote by M(A) the complement of the union of all hyperplanes of A. The complements M(An−1) and M(Bn) of the complexifications of the two arrangements above are the objects of our study. The topology of such complements have been the subject of studies since the early 1970’s. The development started in 1972, when P. Deligne proved that the complement of a complexified arrangement is K(π, 1) when the chambers of the subdivision of Rn induced by the hyperplanes are simplicial cones [7]. 1 With regard to this thesis one result of M. Salvetti from 1987 is of great importance. He proved that the complement of a complexified real hyperplane arrangement is homotopy equivalent to a regular CW-complex [18]. Since the groups Hi(Xi,Xi−1) of the cellular cochain complex of a CW-complex X are free abelian with basis in one-to-one correspondence with the i-cells of X, we call a CW-complex minimal if its number of cells of dimension i equals the rank of the cohomology group Hi(X,Q). Taking the regular CW-complexes, which are based on Salvetti’s work, as a starting point, we derive minimal CW-complexes An−1 and Bn for the complements M(An−1) ⊂ Cn and M(Bn) ⊂ Cn of the complexifications of the two arrangements above. Hence, we deduce CW-complexes which are homotopy equivalent to M(An−1) or M(Bn) and which have a minimal number of cells. In order to decrease the number of cells, discrete Morse Theory provides our basis tool. It was developed by R. Forman in the late 1990’s. Discrete Morse Theory allows to decimate the number of cells of a regular CW-complex without changing its homotopy type. Parallel to our work, a general approach to finding a CW-complex homotopic to the complement of an arrangement using discrete Morse theory was developed in [19]. Our approach is different for the cases studied and leads to a much more explicit description than the statement in [19]. It is well known that the rank of the cohomology groups Hi(M(An−1),Q) and Hi(M(Bn),Q) of the complementsM(An−1) andM(Bn) equals the number of elements of length i in the underlying reflection groups Sn and SB n , respectively [1]. Here, Sn is the symmetric group and SB n is the group of signed permutations, consisting of all bijections ω of the set [±n] := {1, . . . , n,−n, . . . ,−1} onto itself, such that ω(−a) = −ω(a) for all a ∈ [±n]. Indeed, the numbers of cells of the minimal complexes An−1 and Bn are equal to the numbers of elements in Sn and SB n , respectively. The cell-order of a CW-complex X is defined to be the order relation on the cells of X with σ ≤ τ for two cells σ, τ of X if and only if the closure of σ is contained in the closure of τ . The poset of all cells of X ordered in this way is called the face poset of X. A main part of this thesis is devoted to the cell-orders of the minimal CW-complexes. In case of the complex An−1 the face poset turns out to have a concise description. The combinatorics of the face poset of Bn seems to be too complicated to be described through a concise and explicit rule. Thus we formulate a description in terms of mechanisms which allow to construct the cells B with A < B from a given cell A. Even though this description is relatively compact, there 2 is still a lot of combinatorics included that has yet to be discovered.

For any graph G=(V,E) without loops, let C2(G) denote the regular CW-complex obtained from G by attaching to each circuit C of G a disc. We show that if G is the suspension of a flat graph, then C2(G) has an embedding into 4-space. Furthermore, we show that for any graph G in the collection of graphs that can be obtained from K7 and K3,3,1,1 by a series of ΔY- and YΔ-transformations, C2(G) cannot be embedded into 4-space.

In 1968 K. Borsuk asked: Is it true that every finite polyhedron dominates only finitely many different shapes? In some previous paper we have shown that an answer to this question is negative. However, using the localization in homotopy theory, we obtain that every simply-connected polyhedron dominates only finitely many different shapes.

Topologies on base point preserving function spaces are studied making use of the Brown–Booth–Tillotson C -smash product and the topologies on function spaces defined by the classes C of exponentiable spaces. Some conditions on the classes C are obtained for exponential bijections and exponential homeomorphisms in the category of topological spaces with base point. Conditions for exponential homeomorphisms are obtained for CW-complexes whose cellular structure implies precise results.